Question

The area of the region bounded by the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) is

• A. \( \pi ab \) sq units
• B. \( \pi^2 ab \) sq units
• C. \( \pi ab^2 \) sq units
• D. \( \pi a b^2 \) sq units

Hint:

The area of an ellipse is calculated as \( \pi \times a \times b \), where \( a \) is the semi-major axis and \( b \) is the semi-minor axis.

Answer 1

The Correct Option is A

Step 1: General equation of an ellipse.
The general equation of an ellipse with semi-major axis \( a \) and semi-minor axis \( b \) is:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]

Step 2: Formula for the area of an ellipse.
The formula for the area of an ellipse is given by: \[ \text{Area} = \pi \times a \times b \]
where \( a \) is the length of the semi-major axis and \( b \) is the length of the semi-minor axis.

Step 3: Applying the formula.
In this case, the equation of the ellipse is given, and we can directly apply the formula for the area. Substituting the values of \( a \) and \( b \) from the equation:
\[ \text{Area} = \pi \times a \times b \]

Step 4: Conclusion.
Therefore, the area of the region bounded by the ellipse is \( \pi ab \) square units, and the correct answer is option (A).